A histogram is an approximate representation of the
distributionDistribution may refer to:
Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distr ...
of numerical data. It was first introduced by
Karl Pearson (; born Carl Pearson; 27 March 1857 – 27 April 1936) was an English mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of ...
To construct a histogram, the first step is to " bin
" (or "
A bucket is typically a watertight, vertical cylinder
A cylinder (from ) has traditionally been a Solid geometry, three-dimensional solid, one of the most basic of curvilinear geometric shapes. Geometrically, it can be considered as a Prism ( ...
") the range of values—that is, divide the entire range of values into a series of intervals—and then count how many values fall into each interval. The bins are usually specified as consecutive, non-overlapping intervals
of a variable. The bins (intervals) must be adjacent and are often (but not required to be) of equal size.
If the bins are of equal size, a rectangle is erected over the bin with height proportional to the
Frequency is the number of occurrences of a repeating event per unit of time
A unit of time is any particular time
Time is the indefinite continued sequence, progress of existence and event (philosophy), events that occur in an apparen ...
—the number of cases in each bin. A histogram may also be
Normalization or normalisation refers to a process that makes something more normal or regular. Most commonly it refers to:
* Normalization (sociology) or social normalization, the process through which ideas and behaviors that may fall outside of ...
to display "relative" frequencies. It then shows the proportion of cases that fall into each of several
Category, plural categories, may refer to:
Philosophy and general uses
Categorization is the human ability and activity of recognizing shared features or similarities between the elements of the experience of the world (such ...
, with the sum of the heights equaling 1.
However, bins need not be of equal width; in that case, the erected rectangle is defined to have its ''area'' proportional to the frequency of cases in the bin. The vertical axis is then not the frequency but ''frequency density''—the number of cases per unit of the variable on the horizontal axis. Examples of variable bin width are displayed on Census bureau data below.
As the adjacent bins leave no gaps, the rectangles of a histogram touch each other to indicate that the original variable is continuous.
Histograms give a rough sense of the density of the underlying distribution of the data, and often for
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a nu ...
: estimating the
probability density function
and probability density function of a normal distribution .
Image:visualisation_mode_median_mean.svg, 150px, Geometric visualisation of the mode (statistics), mode, median (statistics), median and mean (statistics), mean of an arbitrary probabilit ...
of the underlying variable. The total area of a histogram used for probability density is always normalized to 1. If the length of the intervals on the ''x''-axis are all 1, then a histogram is identical to a relative frequency
A histogram can be thought of as a simplistic
kernel density estimation
Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with ...
, which uses a
Kernel may refer to:
* Kernel (operating system)
In an operating system with a Abstraction layer, layered architecture, the kernel is the lowest level, has complete control of the hardware and is always in memory. In some systems it ...
to smooth frequencies over the bins. This yields a
probability density function, which will in general more accurately reflect distribution of the underlying variable. The density estimate could be plotted as an alternative to the histogram, and is usually drawn as a curve rather than a set of boxes. Histograms are nevertheless preferred in applications, when their statistical properties need to be modeled. The correlated variation of a kernel density estimate is very difficult to describe mathematically, while it is simple for a histogram where each bin varies independently.
An alternative to kernel density estimation is the average shifted histogram,
which is fast to compute and gives a smooth curve estimate of the density without using kernels.
The histogram is one of the seven basic tools of quality control
Histograms are sometimes confused with bar charts. A histogram is used for continuous data
, where the bins represent ranges of data, while a
A bar chart or bar graph is a chart or graph that presents Categorical variable, categorical data with Rectangle, rectangular bars with heights or lengths proportional to the values that they represent. The bars can be plotted vertically or horizo ...
is a plot of
In statistics, a categorical variable is a variable (research), variable that can take on one of a limited, and usually fixed, number of possible values, assigning each individual or other unit of observation to a particular group or nominal catego ...
s. Some authors recommend that bar charts have gaps between the rectangles to clarify the distinction.
This is the data for the histogram to the right, using 500 items:
The words used to describe the patterns in a histogram are: "symmetric", "skewed left" or "right", "unimodal", "bimodal" or "multimodal".
Symmetric-histogram.png, Symmetric, unimodal
It is a good idea to plot the data using several different bin widths to learn more about it. Here is an example on tips given in a restaurant.
Tips-histogram1.png, Tips using a $1 bin width, skewed right, unimodal
Tips-histogram2.png, Tips using a 10c bin width, still skewed right, multimodal with modes at $ and 50c amounts, indicates rounding, also some outliers
U.S. Census Bureau
The United States Census Bureau (USCB), officially the Bureau of the Census, is a principal agency of the U.S. Federal Statistical System, responsible for producing data about the American people and economy. The Census Bureau is part of th ...
found that there were 124 million people who work outside of their homes. Using their data on the time occupied by travel to work, the table below shows the absolute number of people who responded with travel times "at least 30 but less than 35 minutes" is higher than the numbers for the categories above and below it. This is likely due to people rounding their reported journey time. The problem of reporting values as somewhat arbitrarily
is a common phenomenon when collecting data from people.
This histogram shows the number of cases per
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
as the height of each block, so that the area of each block is equal to the number of people in the survey who fall into its category. The area under the curve represents the total number of cases (124 million). This type of histogram shows absolute numbers, with Q in thousands.
This histogram differs from the first only in the
Vertical may refer to:
* Vertical direction
Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, celestial objects ...
scale. The area of each block is the fraction of the total that each category represents, and the total area of all the bars is equal to 1 (the fraction meaning "all"). The curve displayed is a simple density estimate
. This version shows proportions, and is also known as a unit area histogram.
In other words, a histogram represents a frequency distribution by means of rectangles whose widths represent class intervals and whose areas are proportional to the corresponding frequencies: the height of each is the average frequency density for the interval. The intervals are placed together in order to show that the data represented by the histogram, while exclusive, is also contiguous. (E.g., in a histogram it is possible to have two connecting intervals of 10.5–20.5 and 20.5–33.5, but not two connecting intervals of 10.5–20.5 and 22.5–32.5. Empty intervals are represented as empty and not skipped.)
The data used to construct a histogram are generated via a function ''m''''i''
that counts the number of observations that fall into each of the disjoint categories (known as ''bins''). Thus, if we let ''n'' be the total number of observations and ''k'' be the total number of bins, the histogram data ''m''''i''
meet the following conditions:
A cumulative histogram is a mapping that counts the cumulative number of observations in all of the bins up to the specified bin. That is, the cumulative histogram ''M''''i''
of a histogram ''m''''j''
is defined as:
Number of bins and width
There is no "best" number of bins, and different bin sizes can reveal different features of the data. Grouping data is at least as old as Graunt
's work in the 17th century, but no systematic guidelines were given
[ until Sturges' work in 1926.] [
Using wider bins where the density of the underlying data points is low reduces noise due to sampling randomness; using narrower bins where the density is high (so the signal drowns the noise) gives greater precision to the density estimation. Thus varying the bin-width within a histogram can be beneficial. Nonetheless, equal-width bins are widely used.
Some theoreticians have attempted to determine an optimal number of bins, but these methods generally make strong assumptions about the shape of the distribution. Depending on the actual data distribution and the goals of the analysis, different bin widths may be appropriate, so experimentation is usually needed to determine an appropriate width. There are, however, various useful guidelines and rules of thumb.
The number of bins ''k'' can be assigned directly or can be calculated from a suggested bin width ''h'' as:
The braces indicate the ] ceiling function
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ....
which takes the square root of the number of data points in the sample (used by Excel's Analysis Toolpak histograms and many other) and rounds to the next
An integer (from the Latin
Latin (, or , ) is a classical language
A classical language is a language
A language is a structured system of communication
Communication (from Latin ''communicare'', meaning "to share" or "to ....
is derived from a binomial distribution
In probability theory and statistics, the Binomial coefficient, binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' statistical independence, indep ... and implicitly assumes an approximately normal distribution.
Sturges' formula implicitly bases bin sizes on the range of the data, and can perform poorly if , because the number of bins will be small—less than seven—and unlikely to show trends in the data well. On the other extreme, Sturges' formula may overestimate bin width for very large datasets, resulting in oversmoothed histograms. It may also perform poorly if the data are not normally distributed.
When compared to Scott's rule and the Terrell-Scott rule, two other widely accepted formulas for histogram bins, the output of Sturges' formula is closest when .
The Rice Rule is presented as a simple alternative to Sturges' rule.
[Doane DP (1976) Aesthetic frequency classification. American Statistician, 30: 181–183] is a modification of Sturges' formula which attempts to improve its performance with non-normal data.
where is the estimated 3rd-moment- skewness
In probability theory
Probability theory is the branch of concerned with . Although there are several different , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of . Typically these ... of the distribution and
Scott's normal reference rule
Bin width is given by
where is the sample
In statistics, the standard deviation is a measure of the amount of variation or statistical dispersion, dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected v .... Scott's normal reference rule is optimal for random samples of normally distributed data, in the sense that it minimizes the integrated mean squared error of the density estimate.
The Freedman–Diaconis rule gives bin width as:
which is based on the ] interquartile range
250px, Boxplot (with an interquartile range) and a probability density function (pdf) of a Normal Population
In descriptive statistics, the interquartile range (IQR) is a measure of statistical dispersion, which is the spread of the data. The I ..., denoted by IQR. It replaces 3.5σ of Scott's rule with 2 IQR, which is less sensitive than the standard deviation to outliers in data.
Minimizing cross-validation estimated squared error
This approach of minimizing integrated mean squared error from Scott's rule can be generalized beyond normal distributions, by using leave-one out cross validation:
Here, is the number of datapoints in the ''k''th bin, and choosing the value of ''h'' that minimizes ''J'' will minimize integrated mean squared error.
Shimazaki and Shinomoto's choice
The choice is based on minimization of an estimated ''L''2
risk functionIn mathematical optimization
File:Nelder-Mead Simionescu.gif, Nelder-Mead minimum search of Test functions for optimization, Simionescu's function. Simplex vertices are ordered by their values, with 1 having the lowest ( best) value., alt=
where and are mean and biased variance of a histogram with bin-width , and .
Variable bin widths
Rather than choosing evenly spaced bins, for some applications it is preferable to vary the bin width. This avoids bins with low counts. A common case is to choose ''equiprobable bins'', where the number of samples in each bin is expected to be approximately equal. The bins may be chosen according to some known distribution or may be chosen based on the data so that each bin has samples. When plotting the histogram, the ''frequency density'' is used for the dependent axis. While all bins have approximately equal area, the heights of the histogram approximate the density distribution.
For equiprobable bins, the following rule for the number of bins is suggested:
This choice of bins is motivated by maximizing the power of a Pearson chi-squared test testing whether the bins do contain equal numbers of samples. More specifically, for a given confidence interval it is recommended to choose between 1/2 and 1 times the following equation:
Where is the
In probability theory and statistics, the probit function is the quantile function associated with the standard normal distribution. It has applications in data analysis and machine learning, in particular Q–Q plot, exploratory statistical gra ... function. Following this rule for would give between and ; the coefficient of 2 is chosen as an easy-to-remember value from this broad optimum.
A good reason why the number of bins should be proportional to